Coalgebras are a category theoretic concept that enable a study of state
transition systems and their behaviours in a unified
setting [1]}, [2]}. The idea behind a
coalgebraic theory of systems is the following. Given an endofunctor \(\mathcal {F}:\mathcal {C}\rightarrow \mathcal {C}\)  on a concrete category \(\mathcal {C}\) , a coalgebra \(f: X \rightarrow \mathcal {F}X\)  represents a
transition system on the set of states \(X\)  possibly with additional structure.
For example, in \(\mathsf {Set}\) , the category of sets and maps, a coalgebra for
the endofunctor \(\mathcal {F}_d X = 2 \times X^A\)  represents a deterministic automaton with
states \(X\)  and alphabet \(A\) . This is because a coalgebra for \(\mathcal {F}_d\)  can be seen
as a map \(\langle o, \delta \rangle : X \rightarrow 2 \times X^A\)  where \(o: X \rightarrow 2\)  is
the output map indicating whether a given state is accepting and \(\delta : X \rightarrow X^A\)  is the transition map where \(\delta (x)(a)\)  is the next state on reading the
letter \(a\)  in state \(x\) . Here \(2=\lbrace 0,1\rbrace \)  is the set of truth values. Similarly
a coalgebra for the endofunctor \(\mathcal {F}_n(X) = 2 \times (\mathcal {P}_\omega X)^A\)  on
\(\mathsf {Set}\) , where \(\mathcal {P}_\omega X\)  is the finite powerset of \(X\) , represents a
nondeterministic automaton.
